From 85da60b7cff8b120ba3620a65cc9d86792d86940 Mon Sep 17 00:00:00 2001
From: Bowy La Riviere <b.m.lariviere@student.tudelft.nl>
Date: Tue, 30 Mar 2021 10:55:25 +0000
Subject: [PATCH] changes effective mass notation and few small bug fixes
---
src/13_semiconductors.md | 10 +++++-----
1 file changed, 5 insertions(+), 5 deletions(-)
diff --git a/src/13_semiconductors.md b/src/13_semiconductors.md
index 74bd4076..5a1429cf 100644
--- a/src/13_semiconductors.md
+++ b/src/13_semiconductors.md
@@ -58,7 +58,7 @@ Before proceeding further, let us remind ourselves of important band structure p
* Group velocity $v=\hbar^{-1}\partial E(k)/\partial k$.
Descibes how quickly electrons move within the lattice.
-* Effective mass $m_{eff} = \hbar^2\left(d^2 E(k)/dk^2\right)^{-1}$.
+* Effective mass $m^* = \hbar^2\left(d^2 E(k)/dk^2\right)^{-1}$.
Tells us how hard it is to *accelerate* the particles and is related to the curvature of the band.
* Density of states $g(E) = \sum_{\textrm{FS}} (dn/dk) \times (dk/dE)$.
The amount of states per infinitesimal interval of energy at given energy.
@@ -69,7 +69,7 @@ In order to check that everything makes sense, we apply the concepts to the free
$$H = \frac{\hbar^2 k^2}{2m}$$
The velocity is $\hbar^{-1}\partial E(k)/\partial k = \hbar k / m \equiv p/m$.
-The effective mass is $m_{eff} = \hbar^2\left(d^2 E(k)/dk^2\right)^{-1} = m$.
+The effective mass is $m^* = \hbar^2\left(d^2 E(k)/dk^2\right)^{-1} = m$.
So in this simplest case the definitions match the usual expressions.
@@ -81,7 +81,7 @@ For example, both filled and empty bands carry no electric current:
$$
\begin{align}
j = 2e \frac{1}{2\pi} \int_{-\pi/a}^{\pi/a} v(k) dk = 2e \frac{1}{2\pi \hbar} \int_{-\pi/a}^{\pi/a} \frac{dE}{dk} \times dk = \\
-2e \frac{1}{2\pi \hbar} [E(-\pi/a) - E(\pi/a)] = 0.
+2e \frac{1}{2\pi \hbar} [E(\pi/a) - E(-\pi/a)] = 0.
\end{align}
$$
On the other hand, a filled band has an equal number of electrons going forwards and backwards which thus cancel and lead to zero current.
@@ -111,7 +111,7 @@ Naturally, dealing with electrons is more convenient whenever a band is almost e
## Properties of holes
-Let us compare the properties of an electron and a hole obtained by removing that electron.
+Let us compare the properties of an electron with energy $E$ and a hole obtained by removing that electron.
Since removing an electron reduces the total energy of the system, the hole's energy is opposite to that of an electron $E_h = -E$.
The probability for an electron state to be occupied in equilibrium is given by $f(E)$:
@@ -312,7 +312,7 @@ This, however only means that the electron current is opposite of the hole curre
1. Consider the top of the valence band of a semiconductor (see [above](#semiconductors-materials-with-two-bands)). Does an electron near the top of the valence band have a positive or a negative effective mass? Does the electron's energy increase or decrease as $k$ increases from 0? Does the electron have a positive or negative group velocity for $k>0$?
2. Answer the same last 3 questions for a hole in the valence band.
- 3. We now consider an electron in a 2D semiconductor near the bottom of the conduction band described by an energy dispersion $E=E_{G}+\frac{\hbar^2}{2m_{eff}}(k_x^2+k_y^2)$. The electron's velocity is given by $\mathbf{v}=\nabla_\mathbf{k} E/\hbar = \frac{1}{\hbar}(\frac{\partial E}{\partial k_x}\mathbf{\hat{x}} + \frac{\partial E}{\partial k_y}\mathbf{\hat{y}})$. Suppose we turn on a magnetic field $B$ in the $z$-direction. Write down the equation of motion for this electron (neglecting collisions). What is the shape of the motion of the electron? What is the characteristic 'cyclotron' frequency of this motion? What is the direction of the Lorentz force with respect to $\nabla_\mathbf{k} E$?
+ 3. We now consider an electron in a 2D semiconductor near the bottom of the conduction band described by an energy dispersion $E=E_{G}+\frac{\hbar^2}{2m^*}(k_x^2+k_y^2)$. The electron's velocity is given by $\mathbf{v}=\nabla_\mathbf{k} E/\hbar = \frac{1}{\hbar}(\frac{\partial E}{\partial k_x}\mathbf{\hat{x}} + \frac{\partial E}{\partial k_y}\mathbf{\hat{y}})$. Suppose we turn on a magnetic field $B$ in the $z$-direction. Write down the equation of motion for this electron (neglecting collisions). What is the shape of the motion of the electron? What is the characteristic 'cyclotron' frequency of this motion? What is the direction of the Lorentz force with respect to $\nabla_\mathbf{k} E$?
4. Suppose we now consider a hole near the bottom of the conduction band and turn on a magnetic field $B$ in the $z$-direction. Is the direction of the circular motion (i.e., the chirality) of the hole the same as that of the electron? Would the chirality change if we instead consider a hole (or electron) near the top of the valence band?
#### Exercise 2: holes in Drude and tight binding model
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